Steiner Triple Systems Intersecting in Pairwise Disjoint Blocks

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1 Steiner Triple Systems Intersecting in Pairwise Disjoint Blocks Yeow Meng Chee Netorics Pte Ltd 130 Joo Seng Road #05-02 Olivine Building Singapore Submitted: Feb 18, 2003; Accepted: Mar 12, 2004; Published: Apr 2, 2004 Abstract Two Steiner triple systems (X, A) and(x, B) are said to intersect in m pairwise disjoint blocks if A B = m and all blocks in A B are pairwise disjoint. For each v, we completely determine the possible values of m such that there exist two Steiner triple systems of order v intersecting in m pairwise disjoint blocks. 1 Introduction A set system is a pair (X, A), where X is a finite set of points, anda is a set of subsets of X, called blocks. Let K be a set of positive integers. The set K is a set of block sizes for (X, A) if A Kfor every A A. Let (X, A) be a set system, and let G = {G 1,...,G s } be a partition of X into subsets, called groups. The triple (X, G, A) isagroup divisible design (GDD) when every 2-subset of X not contained in a group appears in exactly one block and A G 1 for all A A and G G. WedenoteaGDD(X, G, A) byk-gdd if K is a set of block sizes for (X, A). The group type, orsimplytype of a GDD (X, G, A) is the multiset [ G G G]. When more convenient, we use the exponential notation to describe the type of a GDD: A GDD of type g t 1 1 gs ts is a GDD where there are exactly t i groups of size g i.ifgis not specified in a GDD (X, G, A), it is taken that all groups are of size one: G = {{x} x X}. A {3}-GDD of type 1 v is a Steiner triple system of order v, and is denoted by STS(v). Two GDDs D 1 = (X, G 1, A 1 )andd 1 = (X, G 2, A 2 ) of the same type are said to intersect in m blocks if A 1 A 2 = m. If in addition, the blocks in A 1 A 2 are pairwise disjoint (that is, for any A, A A 1 A 2, A A,wehaveA A = ), then D 1 and D 2 the electronic journal of combinatorics 11 (2004), #R27 1

2 are said to intersect in m pairwise disjoint blocks. Define Int(T )={m two {3}-GDDs of type T intersecting in m blocks}; and Int d (T )={m two {3}-GDDs of type T intersecting in m pairwise disjoint blocks}. The intersection problem and disjoint intersection problem for {3}-GDDs of type T is to determine Int(T ) and Int d (T ), respectively. A pair of GDDs is said to be disjoint if they intersect in no blocks. The intersection problem for Steiner triple systems was completely solved by Lindner and Rosa [6]. The intersection problem for {3}-GDDs of type 2 t has also been solved by Hoffman and Lindner [5]. The case of {3}-GDDs of type g 3, which are equivalent to Latin squares of side g, has been settled by Fu [4]. Butler and Hoffman [2] finally put the intersection problem for {3}-GDDs of type g t to rest with the following result. Let b(g t )=g 2 t(t 1)/6 (the number of blocks in a {3}-GDD of type g t ), and denote by I(g t )={0, 1...,b(g t )}\{b(g t ) 5,b(g t ) 3,b(g t ) 2,b(g t ) 1}. Theorem 1.1 (Butler and Hoffman) Let g and t be positive integers such that t 3, g 2( t 2) 0(mod3), and g(t 1) 0(mod2). Then Int(g t )=I(g t ), except that (i) Int(1 9 )=I(1 9 ) \{5, 8}; (ii) Int(2 4 )=I(2 4 ) \{1, 4}; (iii) Int(3 3 )=I(3 3 ) \{1, 2, 5}; and (iv) Int(4 3 )=I(4 3 ) \{5, 7, 10}. While the intersection problem has received considerable attention, the disjoint intersection problem seems not to be well-studied. The purpose of this paper is to determine completely Int d (1 v ). This is the disjoint intersection problem for STS(v). Hence, we assume throughout this paper that v 1, 3 (mod 6). Note that since 0, 1 Int(1 v ), we also have 0, 1 Int d (1 v ). Define I d (1 v )= { {0, 1,...,(v 1)/3} if v 1 (mod 6), {0, 1,...,v/3} if v 3 (mod 6). Since there can be at most v/3 pairwise disjoint blocks in an STS(v), we have Int d (1 v ) I d (1 v ). So in the remaining of this paper, we shall focus on showing membership of elements of I d (1 v )inint d (1 v ), rather than the other way round. 2 Latin Squares Intersecting in a Transversal A Latin square of side n is an n n array with the property that every row and every column contains every element from {1,...,n} exactly once. A Latin square A of side n is idempotent if A(i, i) =s i for 1 i n. A transversal in a Latin square of side n is a the electronic journal of combinatorics 11 (2004), #R27 2

3 set of n cells, no two from the same row or from the same column, or contain the same entry. The intersection of two Latin squares A and B is the set of cells (i, j) such that A(i, j) =B(i, j). In this section, we show that for n 2, 3, 6, there exists a pair of Latin squares of side n intersecting in a transversal. Two Latin squares A and B of side n are said to be orthogonal if the n 2 ordered pairs (A(i, j),b(i, j)) for 1 i, j n are all distinct. The following result is well-known (see, for example, [1]). Theorem 2.1 There exists a pair of orthogonal idempotent Latin squares of side n for all n 2, 3, 6. Theorem 2.2 Let n 2, 3, 6. Then there exists a pair of Latin squares of side n intersecting in a transversal. Proof. Let A and B be a pair of orthogonal idempotent Latin squares of side n. We claim that A and B intersect in a transversal. Obviously, the transversal formed by the main diagonal of either A or B is in the intersection. It remains to show that for 1 i, j n, i j, wehavea(i, j) B(i, j). Suppose on the contrary that A(i, j) =B(i, j). Then we have (A(i, j), B(i, j)) = (k, k) =(A(k, k), B(k, k)) for some k. This contradicts the fact that A and B are orthogonal. A {3}-GDD of type g 3 can be obtained from a Latin square A of side g as follows. Let R = {r 1,...,r g } and C = {c 1,...,c g } be the set of row and column indices of A, respectively. Let S = {1,...,n} be the set of entries of A. Define X = R C S; G = {R, C, S}; A = {{r i,c j,a(i, j)} 1 i, j g}. Then (X, G, A) isa{3}-gdd of type g 3. From this construction, it is easy to see that Theorem 2.2 implies the following. Corollary 2.1 For g 2, 3, 6, we have g Int d (g 3 ). 3 Wilson-Type Constructions Wilson [7] established the following fundamental construction for GDDs which has significant impact on design theory. the electronic journal of combinatorics 11 (2004), #R27 3

4 Input: Output: Notation: Wilson s Fundamental Construction (master) GDD D =(X, G, A); weight function ω : X Z 0 ; (ingredient) K-GDD D A =(X A, G A, B A )oftype[ω(a) a A], for each block A A,where X A = a A {{a} {1,...,ω(a)}} and G A = {{a} {1...,ω(a)} a A}. K-GDD D =(X, G, A )oftype[ x G ω(x) G G], where X = x X ({x} {1,...,ω(x)}), G = { x G ({x} {1,...,ω(x)}) G G},and A = A A B A. D =WFC(D,ω,{D A A A}). We use Wilson s fundamental construction to produce pairs of disjoint GDDs. Theorem 3.1 Let D =(X, G, A) beagdd.letω : X Z 0 be a weight function. Suppose that for each block A A, there exist a pair of disjoint {k}-gdds of type [ω(a) a A]. Then there exists a pair of disjoint {k}-gdds of type [ x G ω(x) G G]. Proof. For each block A A,letDA 1 =(X A, G A, BA 1 )andd2 A =(X A, G A, BA 2 )beapairof disjoint {k}-gdds of type [ω(a) a A]. Let Di =WFC(D,ω,{Di A A A}), i =1, 2. Then D 1 and D 2 are {k}-gdds of type [ x G ω(x) G G]. We claim that Di =(X, G, A i ), i =1, 2, is a pair of disjoint GDDs. To prove this, we have to establish the following: BA 1 B2 A =, for all A A;and (1) BA 1 BB 2 =, for all A, B A, A B. (2) (1) follows trivially from the fact that (X A, G A, BA 1 )and(x A, G A, BA 2 ) are disjoint, for all A A. To see that (2) holds, note that for A = {a 1,...,a k } and B = {b 1,...,b k },blocks in BA 1 have the form {y 1,...,y k },wherey i =(a i,w) for some w {1,...,ω(a i )}, and blocks in BB 2 have the form {z 1,...,z k },wherez i =(b i,w) for some w {1,...,ω(z i )}. Hence y i = z j is possible only if a i = b j. It follows that {y 1,...,y k } = {z 1,...,z k } only if A = B, establishing (2). Theorem 3.2 (Colbourn, Hoffman, and Rees) Let g, t, and u be positive integers. There exists a {3}-GDD of type g t u 1 if and only if the following conditions are all satisfied: (i) t 3, ort =2and u = g; (ii) u g(t 1); (iii) g(t 1) + u 0(mod2); (iv) gt 0(mod2); the electronic journal of combinatorics 11 (2004), #R27 4

5 (v) g 2( t 2) + gtu 0(mod3). Theorem 3.1 together with the existence of some {3}-GDDs provided by the above result of Colbourn, Hoffman and Rees [3] yields the following. Corollary 3.1 For each type listed in the table below, there exist a pair of disjoint {3}- GDDs. Type Type of Master Weight {3}-GDD Function ω( ) = ω( ) =3 9 t 15 1, 3 t 5 1 ω( ) =3 t 0(mod2), t 4 9 t 21 1, 3 t 7 1 ω( ) =3 t 0(mod2), t 4 9 t 27 1, 3 t 9 1 ω( ) =3 t 0(mod2), t 4 (6s) t (6r) 1, (2s) t (2r) 1 ω( ) =3 t 3, r s(t 1), s 2( t 2) + str 0(mod3) Proof. For each desired pair of disjoint {3}-GDDs, apply Theorem 3.1 with the listed master GDD, weight function, and use as ingredient GDDs a pair of disjoint {3}-GDDs of type 3 3, which exists by Theorem 1.1. The master GDDs all exist by Theorem 3.2. Theorem 3.3 (Filling in Groups) Suppose there exists a pair of disjoint K-GDDs of type [g 1,...,g s ]. Furthermore, for each g i, 1 i s, there exists a pair of L-GDDs of type 1 g i intersecting in m i pairwise disjoint blocks. Then there exists a pair of (K L)-GDDs of type 1 P s g i intersecting in s m i pairwise disjoint blocks. Proof. Let (X, G, A) and(x, G, B) be a pair of disjoint K-GDDs of type [g 1,...,g s ], where G = {G 1,...,G s } and G i = g i,1 i s. Foreachg i,1 i s, let(g i, A i )and (G i, B i )beapairofl-gdds intersecting in m i pairwise disjoint blocks. Define D 1 =(X, A ( s A i )); and D 2 =(X, B ( s B i)). D 1 and D 2 are clearly (K L)-GDDs of type 1 P s g i. We claim that D 1 and D 2 intersect in s m i pairwise disjoint blocks. To see this, note that (i) A B=, since(x, G, A) and(x, G, B) are disjoint GDDs; (ii) for 1 i s, wehavea B i = and B A i =, since any block in A contains at most one element from each group; and (iii) for 1 i, j s, i j, wehavea i B j =, since groups are pairwise disjoint. the electronic journal of combinatorics 11 (2004), #R27 5

6 Hence, the blocks that D 1 and D 2 have in common are precisely the blocks in s (A i B i ). It follows that D 1 and D 2 are the desired pair of GDDs intersecting in s m i pairwise disjoint blocks. For A, B Z, we use the notation A + B to denote the set {a + b a A and b B}. For A i Z, 1 i n, weusethenotation n A i to denote the set {a a n a i A i, for 1 i n}. Corollary 3.2 If there exists a pair of disjoint {3}-GDDs of type g t 1 1 g ts s, then Int d (1 P s g it i ) s t i j=1 Int d (1 g i ). Proof. Apply Theorem 3.3 with GDDs intersecting in the appropriate number of pairwise disjoint blocks. The above results are sufficient to settle the case v 3(mod6). 4 Product Constructions We use the following Singular Direct Product construction. Singular Direct Product Construction Input: positive integers u, w, t; for i =1,...,t,(master)K-GDDs D i =(X i U, G i, A i )oftype u 1 1 w i,whereu is the group of size u in each GDD; (ingredient) K-GDD E =( t X i, {X i 1 i t}, B) of type [w i 1 i t]; (fill-in) K-GDD F =(G, C) oftype1 u. Output: K-GDD D =(X, A )oftype1 u+p t w i,where X = U ( t X i), and A =( t A i) B C. Notation: D =SDP({D i 1 i t}, E, F). Lemma 4.1 Let v 3(mod6)and k 0, 1(mod3), k 3. Then k 1 Int d (1 k(v 1)+1 ) Int d (1 v )+ (Int d (1 v ) \{v/3}). Proof. For 1 i k, letd i =(X i { }, G i, A i )andd i =(X i { }, G i, A i)beapairof {3}-GDDs of type 1 v intersecting in m i pairwise disjoint blocks, where m 1 Int d (1 v )and m i Int d (1 v ) \{v/3} for 2 i k. We may assume that the point is not contained in any blocks in the intersection of D i and D i,2 i k. the electronic journal of combinatorics 11 (2004), #R27 6

7 Let E =( k X i, {X i 1 i k}, B) ande =( k X i, {X i 1 i k}, B ) be a pair of disjoint {3}-GDDs of type (v 1) k, which exists by Theorem 1.1. Let F =({ }, ). Now apply the Singular Direct Product construction to obtain D1 = SDP({D 1,...,D k }, E, F) andd2 =SDP({D 1,...,D k }, E, F). It is easy to see that the intersection of D1 and D2 is k (A i A i), which contains k m i pairwise disjoint blocks. Lemma 4.2 Let s and t be positive integers, t 3. Then Int d (1 6st+1 ) Int d (1 6s+1 ). Proof. For 1 i t, letd i =(X i { }, G i, A i )andd i =(X i { }, G i, A i )beapairof {3}-GDDs of type 1 6s+1 intersecting in m i pairwise disjoint blocks, where m i Int d (1 6s+1 ). We may assume that the point is not contained in any of the intersections. Let E =( t X i, {X i 1 i t}, B) ande =( t X i, {X i 1 i t}, B )beapair of disjoint {3}-GDD of type (6s) t, which exists by Theorem 1.1. Let F =({ }, ). Now apply the Singular Direct Product construction to obtain two {3}-GDDs of type 1 6st+1 D1 =SDP({D i 1 i t}, E, F) andd2 =SDP({D i 1 i t}, E, F). It is easy to see that the intersection of D1 and D2 is t (A i A i), which contains t m i pairwise disjoint blocks. Lemma 4.3 Let r, s, t be positive integers such that t 3, r s(t 1), and s 2( t 2) +str 0 (mod 3). Then Int d (1 6st+6r+1 ) Int d (1 6r+1 )+ Int d (1 6s+1 ). Proof. For 1 i t, letd i =(X i { }, G i, A i )andd i =(X i { }, G i, A i )beapairof {3}-GDDs of type 1 6s+1 intersecting in m i pairwise disjoint blocks, where m i Int d (1 6s+1 ). We may assume that the point is not contained in any of the intersections. Let D t+1 =(X t+1 { }, G i, A t+1 )andd t+1 =(X t+1 { }, G i, A t+1 )beapairof{3}-gdds of type 1 6r+1 intersecting in m t+1 pairwise disjoint blocks, where m t+1 Int d (1 6r+1 ). Let E =( t+1 X i, {X i 1 i t +1}, B) ande =( t+1 X i, {X i 1 i t +1}, B ) be a pair of disjoint {3}-GDDs of type (6s) t (6r) 1, which exists by Corollary 3.1. Let F =({ }, ). Now apply the Singular Direct Product construction to obtain two {3}- GDDs of type 1 6st+6r+1 D1 =SDP({D i 1 i t +1}, E, F) andd2 =SDP({D i 1 i t +1}, E, F). It is easy to see that the intersection of D1 and D 2 is t+1 (A i A i ), which contains t+1 m i pairwise disjoint blocks. the electronic journal of combinatorics 11 (2004), #R27 7

8 5 Direct Constructions In this section, we determine Int d (1 v ) for some small values of v. Lemma 5.1 The following equalities hold: (i) Int d (1 1 )={0}; (ii) Int d (1 3 )={1}; (iii) Int d (1 7 )={0, 1}; (iv) Int d (1 9 )={0, 1, 3}; and (v) for v 1, 3(mod6), 13 v 31, Int d (1 v )=I d (1 v ). 5.1 Proof of Lemma 5.1 (i) and (ii) hold trivially and recall that 0, 1 Int d (1 v ) for v 1, 3(mod6),v 7. For 7 v 31, we have the following table. v Elements Authority in Int d (1 v ) 7 0, 1 Trivial 9 0, 1, 3 Corollary 3.2: Int d (1 9 ) 3 Int d(1 3 ) 13 0, 1 Trivial 15 0, 1, 5 Corollary 3.2: Int d (1 15 ) 5 Int d(1 3 ) 19 0, 1, 2, 3 Lemma 4.2: Int d (1 19 ) 3 Int d(1 7 ) 21 0, 1, 2, 3, 4, 5, 7 Corollary 3.2: Int d (1 21 ) 3 Int d(1 7 ); Corollary 3.2: Int d (1 21 ) Int d (1 9 )+ 4 Int d(1 3 ) 25 0, 1, 2, 3, 4, 5 Lemma 4.1: Int d (1 25 ) Int d (1 9 )+ 2 (Int d(1 9 ) \{3}) 27 0, 1, 2, 3, 4, 5, 6, 7, 9 Corollary 3.2: Int d (1 27 ) 3 Int d(1 9 ) 31 0, 1, 2, 3, 4, 5, 6, 7 Lemma 4.3: Int d (1 31 ) Int d (1 13 )+ 3 Int d(1 7 ) Let S v, v {13, 15, 19, 21, 25, 27, 31} be the STS(v) s given in Appendix A. For an STS(v) S v =(X, B) and a permutation π : X X, π acts on S v by canonical extension as follows: π(s v )={{π(a),π(b),π(c)} {a, b, c} B}. The permutations π listed below is such that S v and π(s v ) intersect in m pairwise disjoint blocks, where m is a number whose membership in Int d (1 v ) has not been established in the table above. the electronic journal of combinatorics 11 (2004), #R27 8

9 v Elements π in Int d (1 v ) 13 2 (0 7 3)(1 5 b 4 9 a 2 c)(6)(8) 3 (0 c 3 4 a 8 2 6)(1 7 b 5)(9) 4 (0 5)(1 c)(2 a 6)(3 b)(4 8)(7)(9) 15 2 (0 8 b e d c 2 a 7 6)(1 5) 3 (0 e 6 c a b d 8) 4 ( e a 3 b 1)(4 c 5 9 d 8) 19 4 (0 a 5)(1 d 8 f c e)(3 7)(b g)(h)(i) 5 ( a 1 c 5 f h 8 b 4 i e 3)(9)(d g) 25 6 (0 o 6 3)(1 4 9 j 5 g k 2 l d)(7 h n e)(8 m)(a c i f b) Any two blocks of an STS(7) intersect in a point. So 2 / Int d (1 7 ). In an STS(9) (X, B), if A, B Band A B =, thenx \ (A B) isablockinb. So 2 / Int d (1 9 ). For each of the remaining possible disjoint intersection size m Int d (1 v ), we exhibit an STS(v) D v, listed in Appendix B, whose intersection with S v contains m pairwise disjoint blocks. v Remaining values T v in Int d (1 v ) 19 6 T T T 3 8 T T T 6 9 T 7 10 T 8 6 Piecing Things Together We first deal with the easier case of v 3(mod6). 6.1 The Case v 3 mod v 3 (mod 18) By Theorem 1.1, there exists a pair of disjoint {3}-GDDs of type Corollary 3.2 then gives Int d (1 39 ) 3 Int d(1 13 ) = 3 {0,...,4} = {0,...,12}. By Theorem 1.1, there also exists a pair of disjoint {3}-GDDs of type Corollary 3.2 now gives Int d (1 39 ) 13 Int d(1 3 )= 13 {1} = {13}. Hence Int d(1 39 )=I d (1 39 ). the electronic journal of combinatorics 11 (2004), #R27 9

10 For v 57, consider a pair of disjoint {3}-GDDs of type 9 t 21 1,wheret =(v 21)/9, existence of which is provided by Corollary 3.1. Now apply Corollary 3.2 to obtain This establishes the following. Int d (1 v ) Int d (1 21 )+ = {0,...,7} + Int d (1 9 ) {0, 1, 3} = {0,...,3t +7} = I d (1 v ). Lemma 6.1 Int d (1 v )=I d (1 v ) for v 3 (mod 18), v v 9 (mod 18) By Theorem 1.1, there exists a pair of disjoint {3}-GDDs of type Corollary 3.2 then gives Int d (1 45 ) 3 Int d(1 15 )= 3 {0,...,5} = {0,...,15} = I d(1 45 ). For v 63, consider a pair of disjoint {3}-GDDs of type 9 t 27 1,wheret =(v 27)/9, existence of which is provided by Corollary 3.1. Now apply Corollary 3.2 to obtain This establishes the following. Int d (1 v ) Int d (1 27 )+ = {0,...,9} + Int d (1 9 ) {0, 1, 3} = {0,...,3t +9} = I d (1 v ). Lemma 6.2 Int d (1 v )=I d (1 v ) for v 9 (mod 18), v v 15 (mod 18) By Corollary 3.1, there exists a pair of disjoint {3}-GDDs of type Corollary 3.2 then gives Int d (1 33 ) Int d (1 15 )+ 6 Int d(1 3 )={0,...,5}+{6} = {6,...,11}. Also, by Lemma 4.1, we have Int d (1 33 ) Int d (1 9 )+ 3 (Int d(1 9 )\{3}) ={0, 1, 3}+ 3 {0, 1} = {0, 1, 2, 3, 4, 5, 6}. Hence we have Int d (1 33 )=I d (1 33 ). the electronic journal of combinatorics 11 (2004), #R27 10

11 For v 51, consider a pair of disjoint {3}-GDDs of type 9 t 15 1,wheret =(v 15)/9, whose existence is provided by Corollary 3.1. Now apply Corollary 3.2 to obtain This establishes the following. Int d (1 v ) Int d (1 15 )+ = {0,...,5} + Int d (1 9 ) {0, 1, 3} = {0,...,3t +5} = I d (1 v ). Lemma 6.3 Int d (1 v )=I d (1 v ) for v 15 (mod 18), v The Case v 1 (mod 6) v 1 (mod 18) Let s =2,t = 3. Then Lemma 4.2 gives Int d (1 37 ) 3 Int d(1 13 )= 3 {0,...,4} = {0,...,12} = I d (1 37 ). For v 55, let s =3,t =(v 1)/18. Now apply Lemma 4.2 to obtain This establishes the following. Int d (1 v ) = Int d (1 19 ) {0,...,6} = {0,...,6t} = I d (1 v ). Lemma 6.4 Int d (1 v )=I d (1 v ) for v 1 (mod 18), v v 7 (mod 18) Let v 43. Let w =(v +2)/3. Then w 3(mod6)andw 15. Now apply Corollary 4.1toobtain Int d (1 v ) Int d (1 w )+(Int d (1 w ) \{w/3})+(int d (1 w )\}w/3}) = {0,...,w/3} + {0,...,w/3 1} + {0,...,w/3 1} = {0,...,w 2} = {0,...,(v 1)/3 1}. the electronic journal of combinatorics 11 (2004), #R27 11

12 To show that (v 1)/3 Int d (1 v ), mimic the proof of Lemma 4.1 but with m 1 = m 2 = m 3 = 0 and replace the ingredient {3}-GDD of type (w 1) 3 with a pair of {3}-GDDs of type (w 1) 3 intersecting in w 1 disjoint blocks, existence of which is provided by Corollary 2.1. This establishes the following. Lemma 6.5 Int d (1 v )=I d (1 v ) for v 7 (mod 18), v v 13 (mod 18) Let s =2andt = 4. Then Lemma 4.2 gives Int d (1 49 ) 4 Int d(1 13 )= 4 {0,...,4} = {0,...,16} = I d (1 49 ). For v 67, let t =(v 13)/18. Then t 3. Now apply Corollary 4.3 with r =2and s = 3 to obtain Int d (1 v )=Int d (1 18t+13 ) Int d (1 13 )+ Int d (1 19 ) = {0,...,4} + {0,...,6} = {0,...,6t +4} = I d (1 v ). This establishes the following. Lemma 6.6 Int d (1 v )=I d (1 v ) for v 13 (mod 18), v Conclusion Lemmas 6.1 to 6.6 shows that Int d (1 v )=I d (1 v ) for all v 1, 3(mod6),v 33. Lemma 5.1 gives Int d (1 v ) for all v 1, 3(mod6),v 31. This combines to give the main result of this paper below. Theorem 7.1 Let v 1, 3(mod6). Then Int d (1 v )=I d (1 v ), except that (i) Int d (1 7 )={0, 1}; and (ii) Int d (1 9 )={0, 1, 3}. If there exists a pair of STS(v) intersecting in t pairwise disjoint blocks, we can consider these t pairwise disjoint blocks as groups of a pair of disjoint {3}-GDDs of type 3 t 1 v t, and vice versa. Hence, Theorem 7.1 is equivalent to the following. Theorem 7.2 Let 3t + r 1, 3(mod6), 3t + r 13. Then there exists a pair of disjoint {3}-GDDs of type 3 t 1 r. the electronic journal of combinatorics 11 (2004), #R27 12

13 References [1] Abel, R. J. R., Brouwer, A. E., Colbourn, C. J., and Dinitz, J. H. Mutually orthogonal Latin squares (MOLS). In The CRC Handbook of Combinatorial Designs, C. J. Colbourn and J. H. Dinitz, Eds. CRC Press, Boca Raton, 1996, ch. II.2, pp [2] Butler, R. A. R., and Hoffman, D. G. Intersections of group divisible triple systems. Ars Combin. 34 (1992), [3] Colbourn, C. J., Hoffman, D. G., and Rees, R. A new class of group divisible designs with block size three. J. Combin. Theory Ser. A 59 (1992), [4] Fu, H. L. On the construcion of certain types of Latin squares having prescribed intersections. PhD thesis, Department of Discrete and Statistical Sciences, Auburn University, Auburn, Alabama, [5] Hoffman, D. G., and Lindner, C. C. The flower intersection problem for Steiner triple systems. Ann. Discrete Math. 34 (1987), [6] Lindner, C. C., and Rosa, A. Steiner triple systems with a prescribed number of triples in common. Canad. J. Math. 27 (1975), [7] Wilson, R. M. An existence theory for pairwise balanced designs I: Composition theorems and morphisms. J. Combin. Theory Ser. A 13 (1971), the electronic journal of combinatorics 11 (2004), #R27 13

14 A Some Small Steiner Triple Systems For space efficiency, we list a block {a, b, c} vertically as a b c. A.1 An STS(13) b3469a a ac578bc95acbbacc9bbac9 A.2 An STS(15) bd3478bc34789c789a789a789a78ab 2468ace569ade65abedcdbeecdbbecdd9ce A.3 An STS(19) abcf 13456abce345678ac345678ad4689bg67acd678bd79fabe9beacdhedg 2789dgfihh9ibgdeffbeh9icg5acedieifghchagf8gidcffhgbhiiifh A.4 An STS(21) aabbccdef af345679bdh345678bcf4679abc678adh678bde79dgabe9acgafieicedhgjg 2ejcbhidgk8egafcikjd9ajkehgi5fjghkicgbkfii9kfjh8ehkcdihdfjbjkfjjgekikh A.5 An STS(25) degin34567bcdefh345679abdfj4689abgik679ceghk679bcegj79abchabcef 27bafjchlkmoma8o9jniglkc8hkoeglmin5elhfdjnodlfimojnldknfimo8migjnkigjn aaaabbbccdddeffhil h9abcefgadgicejmcekdkfglhgjijm mndhmkoibjlohnloofmelknoohmlkn A.6 An STS(27) abdeh abdfg acefj4678acdfkn6789abijp6789abcde79adl 2pgnkfqcmiljolhpecnjiqmokhdgimbpkqonl5be9ogijmqcnfkemloqjlihqomkf8gfhq the electronic journal of combinatorics 11 (2004), #R27 14

15 aaabbbbbccccdeeefffggghhiikl mn9abdghacdehkadeincdhcdefjdfiljghkgimjlmjkjoom opqjkpoigjomplbflmopnlhgnlpeknoqiqphqpnpqmnkpqn A.7 An STS(31) acfhijn34567abcdghjms bdefiln4679abcdefjn678bcdfghmn6789 2plbergdotkmqus89elfkoqnirptuoahjkcugqptmsr5mtgiurhklsqrqjeotiskpuiosm aaaaaabbbbbbccccddddeeeeeeffgg acdfglp79abcfghq9abcdijp9abdehikmachijkncefhnpcdfhjkdfgmfgikfghiorgjjn rnjukqt8odpknusteglhmsnufuqrlntpobsqlrtpjqmltsisrtmnepluopulsmjntuhqto ghhiijkkklllmor qiojolmoqmptqps rpukrorsunrusqt the electronic journal of combinatorics 11 (2004), #R27 15

16 B A Few More Small Steiner Triple Systems B.1 An STS(19) B.1.1 T bcf ag34578abd3459abde479bcf78acd68abd79ace9ce9abaehdg 2d9ebfhci6e9higcf867cfghi5ahegibfihggchfi8idfhdiggedbfieh B.2 An STS(21) B.2.1 T aabbccdef abd345678cfg345789abc4789abd6789bi689cde7abcg9adh9efaeefhjdgihg 2ig7jkhcefeha9bdikj6kiejfdgh5fhkgcjecadfjkbgjhf8hdfijigkcgibikjikekkjh B.3 Two STS(25) s B.3.1 T abcd345689adefj345689abefn4689abekl68abcdhl6789abeg7acdhm9abdf 2johemglkinfi7nbhgclkmo7icjdkmghlo5fcdhmgno9fjegknmijmfldok8olgknegcok aabbbccccddeeffgggi hl9abekachjdefhkdfhkhiijgiijmj inoijnlbimnnfnloejomjmlmhonlok B.3.2 T abfi34569cdegjk345789abcef467abdefh6789aehn89abcdeh7bdfghm9abd 2ndgckljehmo8b7afhniomlgi6mhdlonjk59cimokljkfjcglmonijfokml8ejnliolojh aaabbbccccdefghijl eg9abdefaegkdhkcdidfgiifgjkjlm niockmgibhmnfnmlgnejkmlohnokon B.4 An STS(27) B.4.1 T abcdgk abimn345679abdfgo46789abceh679bdeghm689abcdgk79bdg 2ml7fjehnqoipkcfjhlgedqopn8hcekjlimqp5lpqodgfijainkjfopqeoipjnmlq8hoqn the electronic journal of combinatorics 11 (2004), #R27 16

17 aaaabbbccccddeeeeefffghiijl ik9abcdfj9abdghmadflcfilcfhdhijhkghjkpgkljojnnm pmmkqognlcnefkipbpjqgqmopimeklmnlmlonqhoppqkoqn B.5 Three STS(31) s B.5.1 T abceghjp acdegio345678bcdeghim478abdegjlqs6789abceijp fiklsmdoqtunrnrkbjsqfluhmpt9opqfaknstljur5cmhokfirptutmnhdgkulqsjnec aaaaaabbbbbbcccccddddeeeefffff abdfhmt79acdefkmn9abdghlr9bcdhikpadefgkmcijknpcehinqdfgimfhijiklogknpr glrsqou8lohpgiusriesqoputtfjgroqubonjrpusqlrutujmtpretprqmlntmsrphloqu ggghhhijjklllno jkqiknjmommosqr snusotkputnqtss B.5.2 T abdehil345689abcdfgjn345679abdehmqs4678abfghlno69abceghikr6789 2qfunjmsctgokprs7qoklmeihuprtk8nclfgptjioru59redcimjuptldtjpnumqostajo aaaaaabbbbbccccccddddeeeeffff bcdefhk7abdeghkm9bcdefhin9abcdfgiqacegijmefhikldhknodfhjmtijmnghkrgkno gsrimpl8qipfjursumgkpqotshrlnupsotbkqtrnpujnspofrsquelqorulsqolsmthtrs ggghiijjjkllopp ikoljmlmqnmqprs nqrtkuptuunsqut B.5.3 T abcdeijk bcefgmn34567acdfijlnq4678abcdgioq679bcdefjpr hqgmtsunofplrkahipdoqlutjsre8gb9mhkpturos5plfthjsmnrumnitglkoqsurslp aaaaaabbbbbbccccddddeeeefffgg abdefik79acdefhnabcfghjk9abceghjkacdefhjlcdghkpefgjlmdfpshinpghjmginkl octnjum8slojtqkuedquritomqnripsoubkgqruntifujnrpkirsoemtumrquolsrhlssq ghhhiiijkkllmor nnoqjmomlqmotps tptrkqspptnuuqt the electronic journal of combinatorics 11 (2004), #R27 17